Paranormal subgroup

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed paranormal if for any $g\in G$, there exists $x\in ⟨H,H^g⟩$ such that $⟨H,H^x⟩=⟨H,H^g⟩$.

Here $H^g=g^{-1}Hg$ is a conjugate of $H$, and the angled braces are for the subgroup generated.

Relation with other properties

Stronger properties

• Normal subgroup
• Pronormal subgroup
• Abnormal subgroup

Weaker properties

• Polynormal subgroup: It has been conjectured that for finite groups, the two notions coincide; however this has neither been proved nor disproved.

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
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GAP-codable subgroup property

The following simple GAP code can be used to test if a subgroup is paranormal.

IsParanormal := function(G,H)
	    local K,g,x,flag;
	    for g in Set(G) do
	    	K := Group(Union(H,H^g));
		flag := false;
		for x in K do
		    if K = Group(Union(H,H^x)) then flag:=true; fi;
		 od;
		 if (flag = false) then return false; fi;
		od;
	return true;
end;;

References

• On the arrangement of intermediate subgroups by M. S. Ba and Z. I. Borevich
• On the arrangement of subgroups by Z. I. Borevich, Zap. Nauchn. Semin. tOMI, 94, 5-12 (1979)
• On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin. LOMI, 103, 13-19, 1980
• Testing of subgroups of a finite group for some embedding properties like pronormality by V. I. Mysovskikh